Database Reference
In-Depth Information
Let us show that
i
1
is monic and epic as well. For any two morphisms
g
1
,g
2
:
1
1
→
C
such that
i
1
◦
g
1
=
i
1
◦
g
2
, that is,
i
1
◦
g
1
=
⊥
◦
g
1
=
⊥
=
D
id
C
,
g
1
,
i
1
◦
g
1
i
1
◦
g
2
1
i
1
◦
g
2
=
g
2
,
⊥
and by point
3 of Definition
23
) and hence
i
1
is a monomorphism. For any two
g
1
=[
, we obtain
g
1
=
g
2
(from
{
g
1
}=
=
={
g
2
}
1
k
1
,
⊥
]
and
1
0
1
g
2
=[
⊥
]:
+⊥
→
D
such that
g
1
◦
i
1
=
g
2
◦
g
1
, that is,
g
1
◦
i
1
=[
⊥
]◦
k
2
,
C
k
1
,
1
1
1
id
C
,
⊥
=
k
1
,
⊥
=
k
1
(component
⊥
is eliminated according to point 4 in
1
1
1
Definition
20
)
=
g
2
◦
i
1
=[
k
2
,
⊥
]◦
id
C
,
⊥
=
k
2
,
⊥
=
k
2
, we obtain
k
1
=
k
2
,
thus
g
1
=
g
2
and hence
i
1
is an epimorphism.
0
is isomorphic to
C
and that two isomorphic
objects in a category can be substituted, we can eliminate for complex objects all
their components which are equal to the empty database
Remark
From the fact that
C
⊥
0
and, generally, consider
only the
strictly-complex
objects. Now we obtain the first fundamental property of
DB
category which confirms that it is fundamentally different from
Set
:
⊥
Proposition 5
The morphisms
[
id
C
, id
C
]:
C
C
→
C are epic and monic but not
isomorphic
.
Consequently
,
DB
is not a topos
.
Proof
Let for any two morphisms
g
1
=
h
1
,k
1
:
C
→
C
C
and
g
2
=
h
2
,k
2
:
C
→
C
C
,
[
id
C
, id
C
]◦
g
1
=[
id
C
, id
C
]◦
g
2
:
C
→
C
. Let us show that
g
1
=
g
2
.
In fact,
[
id
C
, id
C
]◦
g
1
=[
id
C
, id
C
]◦
h
1
,k
1
=
id
C
◦
h
1
, id
C
◦
k
1
=
h
1
,k
1
:
C
→
C
and
[
id
C
, id
C
]◦
g
2
=
h
2
,k
2
:
C
→
C
. Consequently,
h
1
,k
1
=
h
2
,k
2
=
=
and, from Definition
23
, there exists a bijection
σ
such that
σ(h
1
)
h
2
,
σ(k
1
)
k
2
with
h
1
=
σ(h
1
)
=
h
2
and
k
1
=
σ(k
1
)
=
k
2
, so that
h
1
=
h
2
and
k
1
=
k
2
and
hence
g
1
=
h
1
,k
1
=
h
2
,k
2
=
g
2
. Consequently,
[
id
C
, id
C
]:
C
C
→
C
is a
monomorphism.
Let for any two morphisms
g
1
:
→
C
and
g
2
:
→
C
,
g
1
◦[
id
C
, id
C
]=
C
C
1
g
2
◦[
id
C
, id
C
]
(the case when
g
1
=
g
2
=⊥
is trivial and hence we consider the
1
). Let us show that
g
1
=
g
2
. In fact,
cases when
g
1
and
g
2
are different from
⊥
g
1
◦[
id
C
, id
C
]=[
g
1
,g
1
]:
C
C
→
C
and
[
g
2
◦
id
C
, id
C
]=[
g
2
,g
2
]:
C
→
C
.
Consequently,
[
g
1
,g
1
]=[
g
2
,g
2
]
and, from point 3 of Definition
23
,
g
1
=
g
2
.
Therefore,
[
id
C
, id
C
]:
C
C
→
C
is an epimorphism. Let us show that
f
=
[
id
C
, id
C
]:
C
C
→
C
is not an isomorphism: If we suppose that it is an iso-
f
−
1
morphism then
f
◦
:
C
→
C
has to be the identity arrow
id
C
.Fromthe
fact that
f
−
1
OP
id
OP
C
, id
O
C
=
=[
id
C
, id
C
]
=
id
C
, id
C
:
C
→
C
C
, so that
f
−
1
◦
=[
id
C
, id
C
]◦
id
C
, id
C
=
id
C
◦
id
C
, id
C
◦
id
C
=
id
C
, id
C
=
f
id
C
(by
point 2 of Definition
23
), i.e., the properties of an isomorphism are not satisfied.
From the fact that in any topos category if a morphism is both epic and monic
then it must also be an isomorphism, we conclude that
DB
is not a topos. More about
this will be considered in Sect.
9.1
dedicated to topological properties of
DB
.
Thus,
DB
is not a topos. However, there is a subset of morphisms for which the
topological properties hold so that if a morphism is both epic and monic then it
must also be an isomorphism: it is the set of all simple arrows, as we will show in
the next corollary, so that the first step is to analyze the topological properties of the